document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1183747>Question 1183747</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the member of orthogonal trajectories which passes through (1, 2) for the family x^2 + 3y^2 = cy? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #814198 by </B><A HREF=/tutors/aboutme.mpl?userid=robertb><B>robertb(5830)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=robertb><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=robertb><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=814198\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> *<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28NOTE%29 ALIGN=MIDDLE  ALT=\"highlight%28NOTE%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:  If you got a question post it here but get it right.  Some people will be glad to help you here.  <BR>\n" );
document.write( "But AVOID or IGNORE ranty, unhelpful tutors.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E2+%2B+3y%5E2+=+cy ALIGN=MIDDLE  ALT=\"x%5E2+%2B+3y%5E2+=+cy\"   DISABLED_style= \"max-width:90%; height:auto;\" > ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=c+=+%28x%5E2+%2B+3y%5E2%29%2Fy ALIGN=MIDDLE  ALT=\"c+=+%28x%5E2+%2B+3y%5E2%29%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "Now by implicit differentiating the given equation, one gets \r<BR>\n" );
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document.write( "2x + 6yy' = cy', which implies that (2x + 6yy')/y' = c.\r<BR>\n" );
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document.write( "Then by equating both expressions for c, one gets\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282x+%2B+6y%2A%28dy%2Fdx%29%29%2F%28dy%2Fdx%29+=+%28x%5E2+%2B+3y%5E2%29%2Fy ALIGN=MIDDLE  ALT=\"%282x+%2B+6y%2A%28dy%2Fdx%29%29%2F%28dy%2Fdx%29+=+%28x%5E2+%2B+3y%5E2%29%2Fy\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2xy+%2B+6y%5E2%2A%28dy%2Fdx%29+=+x%5E2%2A%28dy%2Fdx%29+%2B+3y%5E2%2A%28dy%2Fdx%29 ALIGN=MIDDLE  ALT=\"2xy+%2B+6y%5E2%2A%28dy%2Fdx%29+=+x%5E2%2A%28dy%2Fdx%29+%2B+3y%5E2%2A%28dy%2Fdx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2xy+%2B+3y%5E2%2A%28dy%2Fdx%29+=+x%5E2%2A%28dy%2Fdx%29 ALIGN=MIDDLE  ALT=\"2xy+%2B+3y%5E2%2A%28dy%2Fdx%29+=+x%5E2%2A%28dy%2Fdx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > \r<BR>\n" );
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document.write( "<===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%283y%5E2-x%5E2%29%2A%28dy%2Fdx%29+%2B+2xy+=+0 ALIGN=MIDDLE  ALT=\"%283y%5E2-x%5E2%29%2A%28dy%2Fdx%29+%2B+2xy+=+0\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+-%28%282xy%29%2F%283y%5E2+-+x%5E2%29%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+-%28%282xy%29%2F%283y%5E2+-+x%5E2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Now for orthogonal trajectories, have to solve for <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+=+%283y%5E2+-+x%5E2%29%2F%282xy%29+=+%283%2F%282x%29%29%2Ay+-+x%2F%282y%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx+=+%283y%5E2+-+x%5E2%29%2F%282xy%29+=+%283%2F%282x%29%29%2Ay+-+x%2F%282y%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "<===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dy%2Fdx+-+%283%2F%282x%29%29%2Ay+=+-%28x%2F2%29%2Ay%5E%28-1%29 ALIGN=MIDDLE  ALT=\"dy%2Fdx+-+%283%2F%282x%29%29%2Ay+=+-%28x%2F2%29%2Ay%5E%28-1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which is a Bernoulli D.E. with n = -1.\r<BR>\n" );
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document.write( "To solve this, use the substitution <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v+=+y%5E2 ALIGN=MIDDLE  ALT=\"v+=+y%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >.  Hence yy' = v'/2, so that \r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=dv%2Fdx+-+%283%2Fx%29%2Av+=+-x ALIGN=MIDDLE  ALT=\"dv%2Fdx+-+%283%2Fx%29%2Av+=+-x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "This is a 1st order linear D.E. with integrating factor <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%28int%28-3%2Fx%2Cdx%29%29+=+x%5E%28-3%29 ALIGN=MIDDLE  ALT=\"e%5E%28int%28-3%2Fx%2Cdx%29%29+=+x%5E%28-3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%5E%28-3%29%2A%28dv%2Fdx%29+-+3x%5E%28-4%29%2Av+=+-x%5E%28-2%29 ALIGN=MIDDLE  ALT=\"x%5E%28-3%29%2A%28dv%2Fdx%29+-+3x%5E%28-4%29%2Av+=+-x%5E%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28d%2Fdx%29%28vx%5E%28-3%29%29+=+-x%5E%28-2%29 ALIGN=MIDDLE  ALT=\"%28d%2Fdx%29%28vx%5E%28-3%29%29+=+-x%5E%28-2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=vx%5E%28-3%29+=+-int%28x%5E%28-2%29%2Cdx%29 ALIGN=MIDDLE  ALT=\"vx%5E%28-3%29+=+-int%28x%5E%28-2%29%2Cdx%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=vx%5E%28-3%29+=+x%5E%28-1%29+%2B+k ALIGN=MIDDLE  ALT=\"vx%5E%28-3%29+=+x%5E%28-1%29+%2B+k\"   DISABLED_style= \"max-width:90%; height:auto;\" >  <===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=v+=+x%5E2+%2B+kx%5E3 ALIGN=MIDDLE  ALT=\"v+=+x%5E2+%2B+kx%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > ===> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y%5E2+=+x%5E2+%2B+kx%5E3 ALIGN=MIDDLE  ALT=\"y%5E2+=+x%5E2+%2B+kx%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Solving for k for the member of the family of orthogonal trajectories passing through (1,2), we have <BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E2+=+1%5E2+%2B+k1%5E3 ALIGN=MIDDLE  ALT=\"2%5E2+=+1%5E2+%2B+k1%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, or k = 3.  Therefore, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=highlight%28y%5E2+=+x%5E2+%2B+3x%5E3%29 ALIGN=MIDDLE  ALT=\"highlight%28y%5E2+=+x%5E2+%2B+3x%5E3%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " </SPAN>\n" );
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